Proceedings of Symposia in Pure Mathematics Volume 61 (1997), pp. 1–27
Structure Theory of Semisimple Lie Groups
A. W. Knapp
This article provides a review of the elementary theory of semisimple Lie al- gebras and Lie groups. It is essentially a summary of much of [K3]. The four sections treat complex semisimple Lie algebras, finite-dimensional representations of complex semisimple Lie algebras, compact Lie groups and real forms of complex Lie algebras, and structure theory of noncompact semisimple groups.
1. Complex Semisimple Lie Algebras This section deals with the structure theory of complex semisimple Lie algebras. Some references for this material are [He], [Hu], [J], [K1], [K3], and [V]. Let g be a finite-dimensional Lie algebra. For the moment we shall allow the underlying field to be R or C, but shortly we shall restrict to Lie algebras over C. Semisimple Lie algebras are defined as follows. Let rad g be the sum of all the solvable ideals in g. The sum of two solvable ideals is a solvable ideal [K3, §I.2], and the finite-dimensionality of g makes rad g a solvable ideal. We say that g is semisimple if rad g =0. Within g, let ad X be the linear transformation given by (ad X)Z =[X, Z]. The Killing form is the symmetric bilinear form on g defined by B(X, Y )= Tr(ad X ad Y ). It is invariant in the sense that B([X, Y ],Z)=B(X, [Y,Z]) for all X, Y, Z in g. Theorem 1.1 (Cartan’s criterion for semisimplicity). The Lie algebra g is semisimple if and only if B is nondegenerate. Reference. [K3, Theorem 1.42].
The Lie algebra g is said to be simple if g is nonabelian and g has no proper nonzero ideals. In this case, [g, g]=g. Semisimple Lie algebras and simple Lie algebras are related as in the following theorem. Theorem 1.2. The Lie algebra g is semisimple if and only if g is the direct sum of simple ideals. In this case there are no other simple ideals, the direct sum decomposition is unique up to the order of the summands, and every ideal is the sum of some subset of the simple ideals. Also in this case, [g, g]=g.
1991 Mathematics Subject Classification. Primary 17B20, 20G05, 22E15.
c 1997 A. W. Knapp
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2 A. W. KNAPP
Reference. [K3, Theorem 1.51].
For the remainder of this section, g will always denote a semisimple Lie algebra, and the underlying field will be C. The dual of a vector space V will be denoted V ∗. We discuss root-space decompositions. For our semisimple Lie algebra g, these are decompositions of the form g = h ⊕ gα. α∈∆ Here h is a Cartan subalgebra, defined in any of three equivalent ways [K3, §§II.2–3] as (a) (usual definition) a nilpotent subalgebra h whose normalizer satisfies Ng(h)=h, (b) (constructive definition) the generalized eigenspace for 0 eigenvalue for ad X with X regular (i.e., characteristic polynomial det(λ1 − ad X) is such that the lowest-order nonzero coefficient is nonzero on X), (c) (special definition for g semisimple) a maximal abelian subspace of g in which every ad H, H ∈ h, is diagonable. ∗ The elements α ∈ h are roots, and the gα’s are root spaces, the α’s being defined as the nonzero elements of h∗ such that
gα = {X ∈ g | [H, X]=α(H)X for all H ∈ h} is nonzero. Let ∆ be the set of all roots. This is a finite set. We recall the the classical examples of root-space decompositions [K3, §II.1]. Example 1. g = sl(n, C)={n-by-n complex matrices of trace 0}. The Cartan subalgebra is h = {diagonal matrices in g}. Let 1in(i, j)th place Eij = 0 elsewhere. ∗ Let ei ∈ h be defined by h1 . ei .. = hi. hn Then each H ∈ h satisfies
(ad H)Eij =[H, Eij]=(ei(H) − ej(H))Eij.
So Eij is a simultaneous eigenvector for all ad H, with eigenvalue ei(H) − ej(H). We conclude that (a) h is a Cartan subalgebra, (b) the roots are the (ei − ej)’s for i = j, C (c) gei−ej = Eij.
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Example 2. g = so(2n +1, C)={n-by-n skew-symmetric complex matrices}. For this example one proceeds similarly. Let h = {H ∈ so(2n +1, C) | H = matrix below}. Here H is block diagonal with n 2-by-2 blocks and one 1-by-1 block, 0 ih the jth 2-by-2 block is j , −ihj 0 the 1-by-1 block is just (0).
Let ej(above matrix H)=hj for 1 ≤ j ≤ n. Then
∆={±ei ± ej with i = j}∪{±ek}.
Formulas for the root vectors Eα may be found in [K3, §II.1]. Example 3. g = sp(n, C). This is the Lie algebra of all 2n-by-2n complex matrices X such that 0 I XtJ + JX =0, where J = . −I 0 For this example the Cartan subalgebra h is the set of all matrices H of the form h1 . .. hn H = − h1 . .. −hn
Let ej(above matrix H)=hj for 1 ≤ j ≤ n. Then
∆={±ei ± ej with i = j}∪{±2ek}.
Formulas for the root vectors Eα may again be found in [K3, §II.1]. Example 4. g = so(2n, C). This example is similar to so(2n +1, C) but without the (2n +1)st entry. The set of roots is ∆={±ei ± ej with i = j}. We return to the discussion of general semisimple Lie algebras g. The following are some elementary properties of root-space decompositions:
(a) [gα, gβ] ⊆ gα+β. (b) If α and β are in ∆ ∪{0} and α + β = 0, then B(gα, gβ)=0. (c) If α is in ∆, then B is nonsingular on gα × g−α. (d) If α is in ∆, then so is −α. (e) B|h×h is nondegenerate. Define Hα to be the element of h paired with α. (f) ∆ spans h∗. See [K3, §II.4]. We isolate some deeper properties of root-space decompositions as a theorem.
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Theorem 1.3. Root-space decompositions have the following properties:
(a) If α is in ∆, then dim gα =1. (b) If α is in ∆, then nα is not in ∆ for any integer n ≥ 2. (c) [gα, gβ]=gα+β if α + β =0 . (d) The real subspace h0 of h on which all roots are real is a real form of h, and | | ∗ B h0×h0 is an inner product. Transfer B h0×h0 to the real span h0 of the roots, obtaining ·, · and |·|2. Reference. [K3, §II.4].
Let us now consider root strings. By definition the α string containing β (for α ∈ ∆, β ∈ ∆ ∪{0}) consists of all members of ∆ ∪{0} of the form β + nα with 2β,α n ∈ Z. The n’s in question form an interval with −p ≤ n ≤ q and p − q = . |α|2 Here p − q is a measure of how centered β is in the root string. When p − q is 0, β is exactly in the center. When p − q is large and positive, β is close to the end 2β,α β + qα of the root string. In any event, it follows that is always an integer. |α|2 A consequence of the form of root strings is that if α is in ∆, then the orthogonal ∗ transformation of h0 given by 2ϕ, α s (ϕ)=ϕ − α α |α|2 carries ∆ into itself. The linear transformation sα is called the root reflection in α. An abstract root system is a finite set ∆ of nonzero elements in a real inner product space V such that (a) ∆ spans V , (b) all sα for α ∈ ∆ carry ∆ to itself, 2β,α (c) is an integer whenever α and β are in ∆. |α|2 We say that an abstract root system is reduced if α ∈ ∆ implies 2α/∈ ∆. The relevance of these notions to semisimple Lie algebras is that the root system of a complex semisimple Lie algebra g with respect to a Cartan subalgebra h forms ∗ a reduced abstract root system in h0. See [K3, Theorem 2.42]. There are four kinds of classical reduced root systems: ⊥ n+1 Rn+1 { − | } An has V = i=1 ei in and ∆ = ei ej i = j . The system An arises from sl(n +1, C). n Bn has V = R and ∆ = {±ei ± ej | i = j}∪{±ek}. The system Bn arises from so(2n +1, C). n Cn has V = R and ∆ = {±ei ± ej | i = j}∪{±2ek}. The system Cn arises from sp(n, C). n Dn has V = R and ∆ = {±ei ± ej | i = j}. . The system Dn arises from so(2n, C). We say that an abstract root system ∆ is reducible if∆=∆ ∪ ∆ with ∆ ⊥ ∆ . Otherwise ∆ is irreducible. Theorem 1.4. A semisimple Lie algebra g is simple if and only if the corre- sponding root system ∆ is irreducible.
STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 5
Reference. [K3, Proposition 2.44].
Now we introduce the notions of lexicographic ordering and positive roots for an abstract root system. The construction is as follows. Let ϕ1,...,ϕm be a spanning set for V . Define ϕ to be positive (written ϕ>0) if there exists an index k such that ϕ, ϕi = 0 for 1 ≤ i ≤ k − 1 and ϕ, ϕk > 0. The corresponding lexicographic ordering has ϕ>ψif ϕ − ψ is positive. Fix such an ordering. Call the root α simple if α>0 and if α does not decompose as α = β1 + β2 with β1 and β2 both positive roots.
Theorem 1.5. If l = dim V , then there are l simple roots α1,...,αl, and they are linearly independent. If β is a root and is written as
β = x1α1 + ···+ xlαl, then all the xj have the same sign (if 0 is allowed to be positive or negative), and all the xj are integers. When standard choices are made, the following are the positive roots and simple roots for the classical reduced root systems:
An. The positive roots are the ei − ej with i
2αi,αj Aij = 2 . |αi|
It has the following properties:
(a) Aij is in Z for all i and j, (b) Aii = 2 for all i, (c) Aij ≤ 0 for i = j, (d) Aij = 0 if and only if Aji =0, (e) there exists a diagonal matrix D with positive diagonal entries such that DAD−1 is symmetric positive definite.
6 A. W. KNAPP
An abstract Cartan matrix is a square matrix satisfying properties (a) through (e) as above. To such a matrix we can associate a Dynkin diagram in the standard way. See [K3, §II.5]. We come to the first principal result. Theorem 1.6 (Isomorphism Theorem). Let g and g be complex semisimple Lie algebras with respective Cartan subalgebras h and h and respective root systems ∆ and ∆ . Suppose that a vector space isomorphism ϕ : h → h is given with the property that ϕ carries ∆ one-one onto ∆ . Let the mapping of ∆ to ∆ be denoted α → α . Fix a simple system Π for ∆. For each α in Π, select nonzero root vectors Eα ∈ g for α and Eα ∈ g for α . Then there exists one and only one Lie algebra isomorphism ϕ˜ : g → g such that ϕ˜|h = ϕ and ϕ˜(Eα)=Eα for all α ∈ Π. Reference. [K3, Theorem 2.108].
Examples. 1) An automorphism of the Dynkin diagram yields an automorphism of the Lie algebra. 2) Let ϕ = −1onh. This extends toϕ ˜ : g → g and is used in constructing real forms of g. See Theorem 3.5 and the discussion that follows it. The Weyl group W (∆) of an abstract root system ∆ is defined to be the finite group generated by all root reflections sα for α ∈ ∆. Theorem 1.7. The Weyl group W (∆) of the abstract root system ∆ has the following properties:
(a) Fix a simple system Π={α1,...,αl} for ∆. Then W (∆) is generated by all ∈ ∈ ∈ sαi , αi Π.Ifα is any reduced root, then there exist αj Π and s W (∆) such that sαj = α. (b) If Π and Π are two simple systems for ∆, then there exists one and only one element s ∈ W (∆) such that sΠ=Π . Reference. [K3, Proposition 2.62 and Theorem 2.63].
Briefly conclusion (b) says that W (∆) acts simply transitively on the set of all simple systems. There is a geometric way of formulating this property. Regard V as the dual of its dual V ∗, so that each root has a kernel in V ∗.AWeyl chamber of V ∗ is a connected component of the subset of V ∗ on which every root is nonzero. Each Weyl chamber is an open convex cone, and each root has constant sign on each Weyl chamber. To each simple system corresponds exactly one Weyl chamber, namely the set where each simple root is positive. Conversely each Weyl chamber determines a simple system by this procedure. If the action of W (∆) on V is transferred to an action on V ∗, then (b) says that W (∆) acts simply transitively on the set of Weyl chambers. Dominance is a notion that plays a role with finite-dimensional representations and will be discussed in detail in §2. We call λ ∈ V dominant if λ, α≥0 for all positive roots α. Equivalently λ, α≥0 is to hold for all simple roots α. Theorem 1.8. Fix an abstract root system ∆. (a) If λ is in V , then there exists a simple system Π for which λ is dominant. (b) If λ is in V and if a simple system is specified, then there is some element w of the Weyl group such that wλ is dominant.
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Reference. [K3, Proposition 2.67 and Corollary 2.68].
Here is a handy result that uses dominance in its proof.
Theorem 1.9 (Chevalley’s Lemma). Fix v in V , and let W0 be the subgroup of W (∆) fixing v. Then W0 is generated by the root reflections sα such that v, α =0. Reference. [K3, Proposition 2.72].
Examples. 1) The only reflections sϕ in W (∆) are the root reflections. 2) If an element v of V is fixed by a nontrivial element of W (∆), then some root is orthogonal to v. 3) Any element of order 2 in W (∆) is the product of commuting root reflections. The main correspondence involving complex semisimple Lie algebras relates three classes of objects and isomorphisms, identifying each one with the other two: (1) complex semisimple Lie algebras and isomorphisms of Lie algebras, (2) abstract reduced root systems and invertible linear maps carrying ∆ to ∆ and respecting the integers 2β,α/|α|2, (3) abstract Cartan matrices and equality up to permutation of indices. The passage from (1) to (2) is well defined because any two Cartan subalgebras of g are conjugate via Int g (see [K3, Theorem 2.15]); here Int g is the analytic subgroup of GL(g) with Lie algebra ad g. The passage from (1) to (2) is one-one by the Isomorphism Theorem (Theorem 1.6 above), and it is onto by a result known as the Existence Theorem (see [K3, Theorem 2.111]). The passage (2) to (3) is well defined because any two simple systems are con- jugate via the Weyl group (Theorem 1.7b above). It is one-one by Theorem 1.7a above, and it is onto by a case-by-case construction.
2. Finite-Dimensional Representations of Complex Semisimple Lie Algebras This section deals with finite-dimensional representations of complex semisimple Lie algebras and with the tools needed in their study. Some references for this material are [Hu], [J], [K1], [K2], [K3], and [V]. Except for one segment about the universal enveloping algebra where g will be allowed to be more general, the notation in this section will be as follows: g = complex semisimple Lie algebra h = Cartan subalgebra ∆=∆(g, h) = set of roots
h0 = real form of h where roots are real-valued B = nondegenerate symmetric invariant bilinear form on g that is positive definite on h0 ∈ ∗ Hλ = member of h0 corresponding to λ h0
Here B can be the Killing form, but it does not need to be. In the definition of Hλ, it is understood that ( · )∗ refers to the vector space dual; the correspondence of λ to Hλ is the one induced by B.
8 A. W. KNAPP
A representation ϕ on a complex vector space V is a linear map ϕ : g → End V with ϕ[X, Y ]=ϕ(X)ϕ(Y ) − ϕ(Y )ϕ(X) for all X and Y in g. Isomorphism of representations is called equivalence.An irreducible representation is a representation ϕ on a nonzero space V such that ϕ(g)U ⊆ U fails for all proper nonzero subspaces U. ∗ n Fix such a ϕ.Forλ ∈ h , let Vλ be the set of all v ∈ V with (ϕ(H)−λ(H)1) v =0 for all H ∈ h and some n = n(H, V ). If Vλ is nonzero, Vλ is called a generalized weight space, and λ is called a weight. If dim V is finite-dimensional, V is the direct sum of its generalized weight spaces. This is a generalization of the fact from linear algebra about a linear transformation L on a finite-dimensional V that V is the direct sum of the generalized eigenspaces of L.Ifλ is a weight, then the subspace {v ∈ V | ϕ(H)v = λ(H)v for all H ∈ h} is nonzero and is called the weight space corresponding to λ. A source of finite-dimensional representations of g is group representations. Sup- pose that G is a compact connected Lie group whose Lie algebra g0 has complexi- fication g.Arepresentation ΦofG on a complex vector space V is a continuous group homomorphism Φ : G → GL(V ). If V is finite-dimensional, then Φ is automatically smooth. We can differentiate to get a representation ϕ of g0 on V , and then we can complexify, writing
ϕ(X + iY )=ϕ(X)+iϕ(Y ), to obtain a representation ϕ of g on V . We can obtain some initial examples of this sort with g = sl(n, C) and g = so(n, C). We start with G = SU(n) and G = SO(n) in the two cases. Each of these has a standard representation on Cn, given by the multiplication of matrices and column vectors. For each we can form a contragredient representation on the dual space (Cn)∗. Then we can form tensor products of copies of the standard representation and its dual. Finally we can pass to skew-symmetric tensors, sym- metric tensors, and similar such subspaces. Representations in polynomials arise as symmetric tensors in the tensor product of copies of (Cn)∗. More examples come by starting with the compact connected Lie group G = U(2n) ∩ Sp(n, C), whose complexified Lie algebra is sp(n, C). In this case the standard representation has dimension 2n. In the examples below, we list some representations obtained in this way from G = SU(n) and G = SO(2n + 1). In each case the weights are identified. Also the highest weight, i.e., the largest weight, is identified relative to the lexico- graphic ordering. The Cartan subalgebras and sets of positive roots for sl(n, C) and so(2n +1, C) are the ones in §1. Examples. Let g = sl(n, C). Here the Cartan subalgebra is the diagonal sub- algebra. 1) Let V be the space of polynomials in z1,...,zn and their conjugates homoge- neous of degree N. The action is
(Φ(g)P )(z,z¯)=P (g−1z,g¯−1z¯).
STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 9 n − ≥ ≥ The weights are all expressions j=1(lj kj)ej with all kj 0 and lj 0 and with n j=1(kj + lj)=N. The highest weight is Ne1. 2) Let V be the subspace of holomorphic polynomials in the preceding example. −1 − n The action is Φ(g)(z )=P (g z). The weights are all expressions j=1 kjej with all k ≥ 0 and with n k = N. The highest weight is −Ne . j j=1 j n 3) Let V = lCn with action
Φ(g)(v1 ∧···∧vl)=gv1 ∧···∧gvl. l l The weights are all expressions k=1 ejk , and the highest weight is k=1 ek. Examples. Let g = so(2n+1, C). Here the Cartan subalgebra is block diagonal, containing n 2-by-2 skew-symmetric blocks and one 1-by-1 block whose entry is 0. 1) Let V be the space of all polynomials in x1,...,x2n+1 that are homogeneous −1 of degree N, the action being Φ(g)(x)=P (g x). The weights are all expressions n − ≥ ≥ n j=1(lj kj)ej with all kj 0 and lj 0 and with k0 + j=1(kj + lj)=N. The highest weight is Ne1. 2) Let V = lC2n+1 with l ≤ n and with action as in Example 3for sl(n, C). The weights are all expressions ±e ±···±e with j < ···
10 A. W. KNAPP
(e) each weight space Vµ for ϕλ has dim Vwµ = dim Vµ for all w in the Weyl group W (∆), and each weight µ has |µ|≤|λ| with equality only if µ is in the orbit W (∆)λ.
Reference. [K3, Theorem 5.5]. Later in this section we discuss tools used in the proof. Remark. As a consequence of (e), the Weyl group acts on the weights, pre- serving multiplicities. The extreme weights are those in the orbit of the highest weight.
We can immediately state the second main theorem of the section on represen- tation theory. It concerns complete reducibility. Theorem 2.2. Let ϕ be a complex-linear representation of g on a finite- dimensional complex vector space V . Then V is completely reducible in the sense that there exist invariant subspaces U1,...,Ur of V such that V = U1 ⊕···⊕Ur and such that the restriction of the representation to each Ui is irreducible. Reference. [K3, Theorem 5.29].
The proofs of Theorems 2.1 and 2.2 use three tools: (a) universal enveloping algebra, (b) Casimir element, (c) Verma modules. We review each of these in turn. First we take up the universal enveloping algebra. In the discussion, we shall allow g to be any complex Lie algebra. Let T (g) be the tensor algebra T (g)=C ⊕ g ⊕ (g ⊗ g) ⊕ (g ⊗ g ⊗ g) ⊕···. In T (g), let J be the two-sided ideal generated by all X ⊗ Y − Y ⊗ X − [X, Y ] with X and Y in the space T 1(g) of first-order tensors. The universal enveloping algebra of g is the associative algebra (with identity) given by U(g)=T (g)/J. ∼ Let ι : g → U(g) be the composition ι : g = T 1(g) 3→ T (g) → U(g), so that ι[X, Y ]=ι(X)ι(Y ) − ι(Y )ι(X). The universal enveloping algebra is so named because of the following universal mapping property. Theorem 2.3. Whenever A is a complex associative algebra with identity and π : g → A is a linear mapping such that π(X)π(Y ) − π(Y )π(X)=π[X, Y ] for all X, Y in g, then there exists a unique algebra homomorphism π˜ : U(g) → A such that π˜(1) = 1 and π =˜π ◦ ι. Reference. [K3, Proposition 3.3].
STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 11
Remark. One thinks ofπ ˜ in the theorem as an extension of π from g to all of U(g). This attitude aboutπ ˜ implicitly assumes that ι is one-one, a fact that follows from Theorem 2.5 below.
Theorem 2.4. Representations of g on complex vector spaces stand in one-one correspondence with left U(g) modules in which 1 acts as 1. Reference. [K3, Corollary 3.6]. Remark. The one-one correspondence comes from π → π˜ in the notation of Theorem 2.3.
Theorem 2.5 (Poincar´e-Birkhoff-Witt Theorem). Let {Xi}i∈A be a basis of g, and suppose that a simple ordering has been imposed on the index set A. Then the set of all monomials j1 ··· jn (ιXi1 ) (iXin ) with i1 < ··· Let us now return to our assumption that g is semisimple. We also return to the other notation listed at the start of this section. We shall apply the theorems about U(g) to a representation ϕ of g on a finite-dimensional vector space V .We enumerate the positive roots as β1,...,βm, and we let H1,...,Hl be a basis of h. We use the ordered basis E−β1 ,...,E−βm ,H1,...,Hl,Eβ1 ,...,Eβm . in the Poincar´e-Birkhoff-Witt Theorem. The theorem says that Ep1 ···Epm Hk1 ···Hkl Eq1 ···Eqm −β1 −βm 1 l β1 βm is a basis of U(g). If we apply members of this basis to a nonzero highest weight vector v of V , we get control of a general member of U(g)v. In fact, Eq1 ···Eqm β1 βm ··· k1 ··· kl will act as 0 if q1 + + qm > 0, and H1 Hl will act as a scalar. Thus we have only to sort out the effect of Ep1 ···Epm , and most of the conclusions in −β1 −βm the Theorem of the Highest Weight (Theorem 2.1) follow readily. This completes the discussion of the universal enveloping algebra. The second tool used in the proofs of Theorems 2.1 and 2.2 is the Casimir element. For our complex semisimple Lie algebra g, the Casimir element Ω is the member Ω= B(Xi,Xj)X˜iX˜j i,j of U(g), where {Xi} is a basis of g and {X˜i} is the dual basis relative to B. One shows that Ω is defined independently of the basis {Xi} and is a member of the center Z(g)ofU(g). (See [K3, Proposition 5.24].) 12 A. W. KNAPP { }l Theorem 2.6. Let Ω be the Casimir element. Let Hi i=1 be an orthonormal basis of h0 relative to B, and choose root vectors Eα so that B(Eα,E−α)=1for all roots α. Then l 2 (a) Ω = i=1 Hi + α∈∆ EαE−α. (b) Ω operates by the scalar |λ|2 +2λ, δ = |λ + δ|2 −|δ|2 in an irreducible finite-dimensional representation of g of highest weight λ, where δ is half the sum of the positive roots. (c) the scalar by which Ω operates in an irreducible finite-dimensional represen- tation of g is nonzero if the representation is not trivial. Reference. [K3, Proposition 5.28]. The Casimir element is used in the proof of complete reducibility (Theorem 2.2). The key special case is that V has an irreducible invariant subspace of codimension 1 and dimension > 1. Then ker Ω is the required invariant complement. This completes the discussion of the Casimir element. The third tool used in the proofs of Theorems 2.1 and 2.2 is the theory of Verma modules. Fix a lexicographic ⊕ ∈ ∗ C ordering, and introduce b = h α>0 gα.Forν h , make into a 1-dimensional C ∈ C C U(h) module ν by defining an action of h by H(z)=ν(H)z for z . Make ν ∈ ∗ into a U(b) module by having α>0 gα act by 0. For µ h , define the Verma module V (µ)by V (µ)=U(g) ⊗U(b) Cµ−δ, where δ is half the sum of the positive roots. (The term “−δ” in the definition is the usual convention and has the effect of simplifying calculations with the Weyl group.) Verma modules have the following elementary properties: (a) V (µ) =0, (b) V (µ) is a universal highest weight module for highest weight modules of U(g) with highest weight µ − δ, (c) each weight space of V (µ) is finite-dimensional, (d) V (µ) has a unique irreducible quotient L(µ). (See [K3, §V.3].) The use of Verma modules allows one to prove the hard step of the Theorem of Highest Weight (Theorem 2.1), which is the existence of an irreducible finite- dimensional representation with given highest weight. In fact, if λ is dominant and algebraically integral, then L(λ + δ) is an irreducible representation with highest weight λ, and all that has to be proved is the finite-dimensionality. The topic of the third main theorem on representation theory in this section is characters, which we treat for now as formal exponential sums. We continue with g as a semisimple Lie algebra, h as a Cartan subalgebra, ∆ as the set of roots, and W (∆) as the Weyl group. Introduce a lexicographic ordering, and let α1,...,αl be the simple roots. ∗ We regard the set Zh of functions from h∗ to Z as an abelian group under ∗ Zh λ pointwise addition. We write elements f of as f = λ∈h∗ f(λ)e . The support ∗ of such an f is defined to be the set of λ ∈ h∗ for which f(λ) = 0. Within Zh , let Z[h∗] be the subgroup of all f of finite support. The subgroup Z[h∗] has a natural commutative ring structure, which is determined by eλeµ = eλ+µ. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 13 We introduce a larger ring, Zh∗. Let l + Q = niαi | all ni ≥ 0,ni ∈ Z . i=1 ∗ Then Zh∗ consists of all f ∈ Zh whose support is contained in the union of + ∗ finitely many sets νi − Q with each νi ∈ h . Then we have inclusions ∗ Z[h∗] ⊆ Zh∗⊆Zh . Multiplication in Zh∗ is given by λ µ ν cλe c˜µe = cλc˜µ e . λ∈h∗ µ∈h∗ ν∈h∗ λ+µ=ν If V is a representation of g (not necessarily finite-dimensional), we say that V has a character (for present purposes) if V is the direct sum of its weight spaces ∞ ∈ ∗ under h, i.e., V = µ∈h∗ Vµ, and if dim Vµ < for µ h . In this case the character is µ char(V )= (dim Vµ)e µ∈h∗ ∗ as a member of Zh . This definition is meaningful if V is finite-dimensional or if V is a Verma module. δ − −α Z ∗ The Weyl denominator is the member d = e α∈∆+ (1 e )of [h ]. In this expression, δ is again half the sum of the positive roots. The Kostant partition function P is the function from Q+ to the nonnegative integers that tells the number of ways, apart from order, that a member of Q+ can be written as the sum of positive roots. By convention, P(0) = 1. Define P −γ ∈ Z ∗ K = γ∈Q+ (γ)e h . Lemma. In the ring Zh∗, Ke−δd =1. Hence d−1 exists in Zh∗. Reference. [K3, Lemma 5.72]. Now we come to the third main theorem. Theorem 2.7 (Weyl Character Formula). Let V be an irreducible finite- dimensional representation of the complex semisimple Lie algebra g with highest weight λ. Then char(V )=d−1 (det w)ew(λ+δ). w∈W (∆) Reference. [K3, Theorem 5.75]. 3. Compact Lie Groups and Real Forms of Complex Lie Algebras This section deals with the structure theory of compact Lie groups and with the existence of compact real forms of complex semisimple Lie algebras. Some references for this material are [He], [K1], [K3], and [V]. Throughout this section, g will denote a finite-dimensional complex Lie algebra, and g0 will denote a finite-dimensional real Lie algebra. Let Zg0 be the center of g0. 14 A. W. KNAPP Let Aut g0 be the automorphism group of g0 as a Lie algebra. This is a closed subgroup of GL(g0), hence a Lie subgroup. Its Lie algebra is Der g0. Let Int g0 be the analytic subgroup of Aut g0 with Lie algebra ad g0.IfG is a connected Lie group with Lie algebra g0, then Ad(G) is an analytic subgroup of GL(g0) with Lie algebra ad g0, hence equals Int g0. Thus Int g0 provides a way of forming Ad(G) without using a particular G. It is the group of inner automorphisms of G or g0. We begin with a discussion of real forms. If we regard g as a real Lie algebra, then a real Lie subalgebra g0 such that g = g0 ⊕ ig0 as vector spaces is called a real form of g. To a real form g0 of g is associated a conjugation of g, which is the R linear map that is 1 on g0 and −1onig0. This is an automorphism of g as a real Lie algebra. If g0 is given, then g0 is a real form of its complexification g = g0 ⊗R C = g0 ⊕ ig0.Ifg0 is a real form of g, then g0 is semisimple if and only if g is semisimple, as a consequence of Cartan’s criterion for semisimplicity (Theorem 1.1). Examples. 1) sl(n, R), su(n), and su(p, q) are real forms of sl(n, C). Here su(n) is the Lie algebra of n-by-n skew-Hermitian matrices of trace 0, and su(p, q) consists of AB matrices of trace 0 in which A and C are skew-Hermitian. B∗ C 2) so(n) is a real form of so(n, C). Here so(n) is the Lie algebra of n-by-n real skew-symmetric matrices. 3) so(p, q) is isomorphic to a real form of so(p + q, C) under conjugation by the 10 AB block diagonal matrix . Here so(p, q) consists of real matrices 0 i Bt C in which Aand C are skew-symmetric. When we complexify and then conjugate 10 by , we obtain so(p + q, C). 0 i 4) sp(n, R) and sp(n, C) ∩ u(2n) are real forms of sp(n, C). The Lie algebra g0 is said to be reductive if to each ideal a0 in g0 corresponds an ideal b0 in g0 with g0 = a0 ⊕ b0. ⊕ Theorem 3.1. The Lie algebra g0 is reductive if and only if g0 =[g0, g0] Zg0 with [g0, g0] semisimple and Zg0 abelian. Reference. [K3, Corollary 1.53]. Now we consider the Lie algebra of a compact Lie group. Theorem 3.2. If G is a compact Lie group and g0 is its Lie algebra, then (a) Int g0 is compact. (b) g0 is reductive. (c) the Killing form of g0 is negative semidefinite. Furthermore let ZG be the center of G, and let Gss be the analytic subgroup of G with Lie algebra [g0, g0]. Then (d) Gss has finite center. (e) (ZG)0 and Gss are closed subgroups. (f) G is the commuting product G =(ZG)0Gss. Reference. [K3, §IV.4]. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 15 Remarks. Conclusions (b) and (c) use the existence of a G invariant inner product on g0, which is constructed using Haar measure on G. Conclusion (d) uses that G may be regarded as a Lie group of matrices; this fact is a consequence of the Peter-Weyl Theorem, which we do not review. See [K3, §IV.3]. Lemma. If g0 is semisimple, then Der g0 =adg0. Hence Int g0 = (Aut g0)0, and Int g0 is a closed subgroup of GL(g0). Reference. [K3, Proposition 1.98]. Remark. Since Int g0 is the group of inner automorphisms of g0 and since Int g0 has Lie algebra ad g0, it is helpful to think of this lemma as saying that every derivation is inner. Theorem 3.3. If the Killing form of g0 is negative definite, then Int g0 is compact. Reference. [K3, Proposition 4.27]. Next we discuss compact real forms. Theorem 3.4. If g0 is semisimple, then the following conditions are equivalent: (a) g0 is the Lie algebra of some compact Lie group. (b) Int g0 is compact. (c) the Killing form of g0 is negative definite. Proof. If G is compact connected with Lie algebra g0, then Ad(G) is compact; hence (a) implies (b). Conversely if (b) holds, then Int g0 is a compact Lie group with Lie algebra ad g0. Since g0 is semisimple, ad g0 is isomorphic to g0; thus (b) implies (a). If (b) holds, then the Killing form is negative semidefinite by Theo- rem 3.2, and it must be negative definite by Cartan’s criterion for semisimplicity (Theorem 1.1). Thus (b) implies (c). Conversely (c) implies (b) by Theorem 3.3. Let g be semisimple. A real form g0 of g is said to be compact if the equivalent conditions of Theorem 3.4 hold. Here are some examples. Examples. su(n) is a compact real form of sl(n, C), so(n) is a compact real form of so(n, C), and sp(n, C) ∩ u(2n) is a compact real form of sp(n, C). Theorem 3.5. Each complex semisimple Lie algebra has a compact real form. Reference. [K3, Theorem 6.11]. This result is fundamental. The first step in the proof is to extend the vector space isomorphism ϕ = −1ofh to an automorphismϕ ˜ of g, using the Isomorphism Theorem (Theorem 1.6). Thenϕ ˜ is used to adjust the structural constants to produce a real form for which the Killing form is negative definite. Application of Theorem 3.4 completes the argument. The next topic is maximal tori. The setting is that G is a compact connected Lie group, g0 is its Lie algebra, g is the complexification of g0, and B is the negative of any Ad(G) invariant inner product on g0. The maximal tori in G are defined to be the subgroups maximal with respect to the property of being compact connected abelian. The theorem below lists the first facts about maximal tori. 16 A. W. KNAPP Theorem 3.6. If G is a compact connected Lie group, then (a) the maximal tori in G are exactly the analytic subgroups corresponding to the maximal abelian subalgebras of g0. (b) any two maximal abelian subalgebras of g0 are conjugate via Ad(G) and hence any two maximal tori in G are conjugate via G. Reference. [K3, Proposition 4.30 and Theorem 4.34]. Here are some standard examples of maximal tori. Examples. 1) Let G = SU(n), the special unitary group. The complexified Lie algebra is g = sl(n, C). A maximal torus, its Lie algebra, and its complexified Lie algebra are T = diag(eiθ1 ,...,eiθn ) t0 = diag(iθ1,...,iθn) t = standard Cartan subalgebra of sl(n, C). 2) Let G = SO(2n + 1), the rotation group. The complexified Lie algebra is g = so(2n +1, C). A maximal torus and its complexified Lie algebra are cos θ sin θ T from 2-by-2 blocks j j and a − sin θj cos θj single 1-by-1 block (1) t = standard Cartan subalgebra of so(2n +1, C). 3) Let G = Sp (n, C) ∩ U(2n). Here Sp(n, C)={x ∈ GL(2n, C) | xtJx = J}, 0 I C where J = −I 0 as earlier. The complexified Lie algebra of G is g = sp(n, ). A maximal torus and its complexified Lie algebra are − − T = diag(eiθ1 ,...,eiθn ,e iθ1 ,...,e iθn ) t = standard Cartan subalgebra of sp(n, C). 4) Let G = SO(2n), the rotation group. The complexified Lie algebra is g = so(2n, C). cos θ sin θ T from 2-by-2 blocks j j − sin θj cos θj t = standard Cartan subalgebra of so(2n, C). The theory of Cartan subalgebras for the complex semisimple case extends to a complex reductive Lie algebras g by just saying that the center of g is to be adjoined to a Cartan subalgebra of the semisimple part of g. Now let us extend the theory of Cartan subalgebras from the complex reductive case to the real reductive case. If g0 is a real reductive Lie algebra, we call a Lie subalgebra of g0 a Cartan subalgebra if its complexification is a Cartan C subalgebra of g =(g0) . Using condition (c) in the definition of Cartan subalgebra for the complex semisimple Lie algebra, we readily see that if g0 is the Lie algebra of STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 17 a compact connected Lie group G and if t0 is a maximal abelian subspace of g0, then t0 is a Cartan subalgebra. In this setting, we can form a root-space decomposition g = t ⊕ gα. α∈∆ Here g = Zg ⊕ [g, g], t = Zg ⊕ (t ∩ [g, g]), and the root spaces gα lie in [g, g]. Moreover, each root is the complexified differential of a multiplicative character ξα of the maximal torus T that corresponds to t0, with Ad(t)X = ξα(t)X for X ∈ gα. The next results concern centralizers of tori. These results give the main control over connectedness of subgroups of semisimple and reductive groups. Theorem 3.7. If G is a compact connected Lie group and T is a maximal torus, then each element of G is conjugate to a member of T . Reference. [K3, Theorem 4.36]. This is a deep theorem. For SU(n), it just amounts to the Spectral Theorem, but it becomes progressively more complicated for more complicated G. We list three immediate consequences. Corollary. (a) Every element of a compact connected Lie group G lies in some maximal torus. (b) The center ZG of a compact connected Lie group lies in every maximal torus. (c) For any compact connected Lie group G, the exponential map is onto G. With a supplementary argument and Theorem 3.7, we obtain Theorem 3.8. Let G be a compact connected Lie group, and let S be a torus of G.Ifg in G centralizes S, then there is a torus S in G containing both S and g. Reference. [K3, Theorem 4.50]. This theorem is normally applied in either of the two forms in the following corollary. Corollary. (a) In a compact connected Lie group, the centralizer of a torus is connected. (b) A maximal torus in a compact connected Lie group is equal to its own cen- tralizer. Let us introduce Weyl groups in this context. The notation is unchanged: G is compact connected, g0 is the Lie algebra of G, g is the complexification, T is a maximal torus, t0 is the Lie algebra of T , t is the complexification, ∆(g, t)isthe set of roots, and B is the negative of a G invariant inner product on g0. Define ∗ tR = it0. Roots are real on tR, hence are in tR. The form B, when extended to be ∗ complex bilinear, is positive definite on tR, yielding an inner product ·, · on tR. 2λ, α Let the root reflection s be defined on t∗ by s (λ)=λ − α. The Weyl α R α |α|2 group W (∆(g, t)) is the group generated by all sα for α ∈ ∆(g, t). This is a finite group. 18 A. W. KNAPP We define W (G, T ) as the quotient of normalizer by centralizer W (G, T )=NG(T )/ZG(T )=NG(T )/T. This also is a finite group. It follows from Theorems 3.7 and 3.6b that the conjugacy classes in G are parametrized by T/W(G, T ). (See [K3, Proposition 4.53].) ∗ Theorem 3.9. The group W (G, T ), when considered as acting on tR, coincides with W (∆(g, t)). Reference. [K3, Theorem 4.54]. Continuing with notation as above, we work with two notions of integrality. It is easy to see that the following two conditions on a member λ of t∗ are equivalent: (1) Whenever H ∈ t0 satisfies exp H = 1, then λ(H)isin2πiZ. λ(H) (2) There is a multiplicative character ξλ of T with ξλ(exp H)=e for all H ∈ t0. When (1) and (2) hold, λ is said to be analytically integral. As before, we say 2λ, α that λ is algebraically integral if is in Z for all α ∈ ∆(g, t). |α|2 Theorem 3.10. Analytic and algebraic integrality have the following eight properties: (a) Weights of finite-dimensional representations of G are analytically integral. In particular, every root is analytically integral. (b) Analytically integral implies algebraically integral. ∗ (c) Fix a simple system of roots {α1,...,αl}. Then λ ∈ t is algebraically 2 integral if and only if 2λ, αi/|αi| is in Z for each simple root αi. (d) If G˜ is a finite covering group of G, then the index of the group of analytically integral forms for G in the group of analytically integral forms for G˜ equals the order of the kernel of the covering homomorphism G˜ → G. ∗ (e) The subgroup of Z combinations of roots in tR is contained in the lattice of analytically integral forms, which in turn is contained in the subgroup of algebraically integral forms. If G is semisimple, all three subgroups are lattices. (f) If G is semisimple, then the index of the lattice of Z combinations of roots in the lattice of algebraically integral forms is exactly the determinant of the Cartan matrix. (g) If G is semisimple and ZG is trivial, then every analytically integral form is a Z combination of roots. (h) If G is simply connected and semisimple, then algebraically integral implies analytically integral. Reference. [K3, §§IV.7 and V.8]. Remarks. In the semisimple case, conclusion (e) identifies containments among ∗ three lattices in tR, and (f) says that the index of the smallest in the largest is the determinant of the Cartan matrix. Conclusions (g) and (h) give circumstances under which the middle lattice is equal to the smallest or the largest. The proof of (h) uses the existence result in the Theorem of the Highest Weight. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 19 Theorem 3.11 (Weyl’s Theorem). If G is a compact semisimple Lie group, then the fundamental group of G is finite. Consequently the universal covering group of G is compact. Reference. [K3, Theorem 4.69]. Combining Weyl’s Theorem with Theorem 3.10, we obtain the following conse- quence. Corollary. In a compact semisimple Lie group G, (a) the order of the fundamental group of G equals the index of the group of analytically integral forms for G in the group of algebraically integral forms. (b) if G is simply connected, then the order of the center ZG of G equals the determinant of the Cartan matrix. Let us now rephrase the results about representations of complex semisimple Lie algebras as results about compact connected Lie groups. (See [K3, §V.8].) Theorem 3.12 (Theorem of the Highest Weight). Let G be a compact con- nected Lie group with complexified Lie algebra g, let T be a maximal torus with complexified Lie algebra t, and let ∆+(g, t) be a positive system for the roots. Apart from equivalence the irreducible finite-dimensional representations Φ of G stand in one-one correspondence with the dominant analytically integral linear functionals λ on t, the correspondence being that λ is the highest weight of Φ. In the context of representations of the compact connected group G, we can λ regard characters char(V )= (dim Vλ)e as functions on t0. The algebraic theory gives d char(V )= (det w)ew(λ+δ) w∈∆(g,t) in Z[t∗] for the semisimple case. We can pass from the algebraic result in Z[t∗] to the group case for G semisimple by using the evaluation homormorphism at each point of t0 and addressing analytic integrality. Then we can extend the group result to general compact connected G. One shows that the element δ ∈ t∗ (half the sum of the positive roots) has 2 2δ, αi/|αi| = 1 for simple αi, hence is algebraically integral. Nevertheless, δ is not always analytically integral; it is not analytically integral in SO(3), for example. A sufficient compensation for this failure is that δ−wδ is always analytically integral for all w. Consequently we are able to obtain the following group version of the Weyl Character Formula. Theorem 3.13 (Weyl Character Formula). Let G be a compact connected Lie group, let T be a maximal torus, let ∆+ =∆+(g, t) be a positive system for the ∗ roots, and let λ ∈ t be analytically integral and dominant. Then the character χλ of the irreducible finite-dimensional representation of G with highest weight λ is given by w ∈W (det w)ξw(λ+δ)−δ(t) χλ = − α∈∆+ (1 ξ−α(t)) at every t ∈ T where no ξα takes the value 1 on t.IfG is simply connected, then this formula can be rewritten as w∈ W (det w)ξw(λ+δ)(t) χλ = − . ξδ(t) α∈∆+ (1 ξ−α(t)) 20 A. W. KNAPP Before concluding the treatment of compact groups, let us mention that much of the theory for compact connected Lie groups can be obtained directly, without first addressing complex semisimple Lie algebras. Weyl carried out such a program, using integration as the tool. Here is the formula that Weyl used. Theorem 3.14 (Weyl Integration Formula). Let T be a maximal torus of the compact connected Lie group G, and let invariant measures on G, T , and G/T be normalized so that f(x) dx = f(xt) dt d(xT ) G G/T T for all continuous f on G. Then every Borel function F ≥ 0 on G has 1 −1 | |2 F (x) dx = | | F (gtg ) d(gT) D(t) dt, G W (G, T ) T G/T where 2 −1 2 |D(t)| = |1 − ξα(t )| . α∈∆+ Reference. [K3, Theorem 8.60]. 4. Structure Theory of Noncompact Semisimple Groups This section deals with the structure theory of noncompact semisimple Lie groups and with the definition and first properties of reductive Lie groups. Some references for this material are [He], [K1], [K3], and [W]. The theory begins with the development of Cartan involutions. Let g0 be a real semisimple Lie algebra, and let B be the Killing form. (Later we shall allow other forms in place of the Killing form.) A source of many examples of real semisimple Lie algebras is as follows. Theorem 4.1. If g0 is a real Lie algebra of real or complex or quaternion matrices closed under conjugate transpose, then g0 is reductive. If also Zg0 =0, then g0 is semisimple. Reference. [K3, Proposition 1.56]. Examples. The following examples are classical Lie algebras that satisfy the hypotheses of Theorem 4.1 for all n, p, and q. For appropriate values of n, p, and q, these examples are semisimple. ∼ 1) Compact Lie algebras: su(n), so(n), and sp(n, C) ∩ u(2n) = sp(n). 2) Complex Lie algebras: sl(n, C), so(n, C), and sp(n, C). 3) Other Lie algebras: sl(n, R), sl(n, H), sp(n, R), so(p, q), su(p, q), sp(p, q), and so∗(2n). Here sl(n, H) refers to quaternion matrices for which the real part of the trace is 0, and sp(p, q) refers to quaternion matrices preserving a Hermitian form of signature (p, q). An involution θ of g0 (understood to respect brackets) such that the symmetric bilinear form Bθ(X, Y )=−B(X, θY ) STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 21 is positive definite is called a Cartan involution of g0. Correspondingly there is a Cartan decomposition of g0 given by g0 = k0 ⊕ p0. The subspaces k0 and p0 are understood to be the +1 and −1 eigenspaces of θ; they satisfy the bracket relations [k0, k0] ⊆ k0, [k0, p0] ⊆ p0, [p0, p0] ⊆ k0. Moreover, B is negative on k0, B is positive on p0, and B(k0, p0)=0. Examples. 1) If g0 is as in the list of examples above, then θ can be taken to be negative conjugate transpose. 2) Let g be a complex semisimple Lie algebra, let u0 be a compact real form of g, and let τ be the corresponding conjugation of g.Ifg is regarded as a real Lie algebra, then τ is a Cartan involution of g. The main tool for handling Cartan involutions is Theorem 4.2 below. This is a result of Berger that improves on the original result of Cartan. Theorem 4.2. Let θ be a Cartan involution of g0, and let σ be any involution. −1 Then there exists ϕ in Int g0 such that ϕθϕ commutes with σ. Reference. [K3, Theorem 6.16]. Corollary. (a) g0 has a Cartan involution. (b) Any two Cartan involutions of g0 are conjugate via Int g0. (c) If g is a complex semisimple Lie algebra, then any two compact real forms of g are conjugate via Int g. (d) If g is a complex semisimple Lie algebra and is considered as a real Lie algebra, then the only Cartan involutions of g are the conjugations with respect to the compact real forms of g. Reference. [K3, §VI.2]. Sketch of proof. For (a), Theorem 4.2 is applied to g made real, using θ from a compact real form and σ from conjugation of g with respect to g0. Conclusion (b) is immediate, and (c) is a special case of (b). Conclusion (d) follows from (b) and the fact that such a conjugation exists (Theorem 3.5). If g0 = k0 ⊕ p0 is a Cartan decomposition of g0, then k0 ⊕ ip0 is a compact real C form of g =(g0) . Conversely Theorem 3.3 shows that if h0 and q0 are the +1 and −1 eigenspaces of an involution σ, then σ is a Cartan involution if the real form C h0 ⊕ iq0 of g =(g0) is compact. These considerations allow B to be generalized a little. Fix an involution θ of g0, and let g0 = k0 ⊕p0 be the eigenspace decomposition relative to θ. We suppose that B is any nondegenerate symmetric invariant bilinear form on g0 with B(θX, θY )= B(X, Y ) such that Bθ(X, Y )=−B(X, θY ) is positive definite. Then B is negative definite on k0 ⊕ ip0, and it follows that k0 ⊕ ip0 is compact. Consequently θ is a Cartan involution. In this setting we allow B to be used in place of the Killing form. 22 A. W. KNAPP Notice in this case that B is negative definite on a maximal abelian subspace of k0 ⊕ ip0, hence positive definite on the real subspace of a Cartan subalgebra of C C (g0) where roots are real-valued. Therefore B has the correct “sign” on (g0) for the theory of complex semisimple Lie algebras to be applicable. By a semisimple Lie group, we mean a connected Lie group whose Lie algebra is semisimple. The next theorem gives the global Cartan decomposition of a semisimple Lie group. Theorem 4.3. Let G be a semisimple Lie group, let θ be a Cartan involution of its Lie algebra g0, let g0 = k0 ⊕ p0 be the corresponding Cartan decomposition, and let K be the analytic subgroup of G with Lie algebra k0. Then (a) there exists a Lie group automorphism Θ of G with differential θ, and Θ has Θ2 =1. (b) the subgroup of G fixed by Θ is K. (c) the mapping K × p0 → G given by (k, X) → k exp X is a diffeomorphism onto. (d) K is closed. (e) K contains the center Z of G. (f) K is compact if and only if Z is finite. (g) when Z is finite, K is a maximal compact subgroup of G. Reference. [K3, Theorem 6.31]. Example. When G is an analytic group of matrices and θ is negative conju- gate transpose, Θ is conjugate transpose inverse. The content of (c) is that G is stable under the polar decomposition of matrices. Thus (c) of the theorem may be regarded as a generalization of the polar decomposition to all semisimple Lie groups. This completes the discussion of Cartan involutions. For most of the remainder of this section, we shall use the following notation. Let G be a semisimple Lie group, let g0 be its Lie algebra, let g be the complexification of g0, let θ be a Cartan involution of g0, and let g0 = k0 ⊕ p0 be the corresponding Cartan decomposition. Let B as above be a θ invariant nondegenerate symmetric bilinear form on g0 such that Bθ is positive definite. The next topic will be restricted roots and the Iwasawa decomposition. Let a0 ∈ ∗ be a maximal abelian subspace of p0. Restricted roots are the nonzero λ a0 such that the space (g0)λ defined as {X ∈ g0 | (ad H)X = λ(H)X for all H ∈ a0} is nonzero. Let Σ be the set of restricted roots. Define m0 = Zk0 (a0). Restricted roots and the corresponding restricted-root spaces have the following elementary properties: ⊕ ⊕ (a) g0 = a0 m0 λ∈Σ(g0)λ, (b) [(g0)λ, (g0)µ] ⊆ (g0)λ+µ, (c) θ(g0)λ =(g0)−λ, ∗ (d) Σ is a root system in a0. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 23 ∗ Introduce a lexicographic ordering in a0, and define Σ+ = {positive restricted roots} n0 = (g0)λ. λ∈Σ+ The subspace n0 of g0 is a nilpotent Lie subalgebra. Theorem 4.4 (Iwasawa decomposition of Lie algebra). The semisimple Lie algebra g0 is a vector-space direct sum g0 = k0 ⊕ a0 ⊕ n0. Here a0 is abelian, n0 is nilpotent, a0⊕n0 is a solvable Lie subalgebra of g0, and a0⊕n0 has [a0⊕n0, a0⊕n0]= n0. Reference. [K3, Proposition 6.43]. Theorem 4.5 (Iwasawa decomposition of Lie group). Let G be a semisimple group, let g0 = k0 ⊕a0 ⊕n0 be an Iwasawa decomposition of the Lie algebra g0 of G, and let A and N be the analytic subgroups of G with Lie algebras a and n. Then the multiplication map K × A × N → G given by (k, a, n) → kan is a diffeomorphism onto. The groups A and N are simply connected. Reference. [K3, Theorem 6.46]. Roots and restricted roots are related to each other. If t0 is a maximal abelian subspace of g0, then h0 = a0 ⊕ t0 is a Cartan subalgebra of g0 (see [K3, Proposition 6.47]). Roots are real-valued on a0 and imaginary-valued on t0. The nonzero restrictions to a0 of the roots turn out to be the restricted roots (see [K3, §VI.4]). Roots and restricted roots can be ordered compatibly by taking a0 before it0. The next theorem describes the effect of altering the choices that have been made in obtaining the Iwasawa decomposition. Theorem 4.6. (a) If a0 and a0 are two maximal abelian subspaces of p0, then there is a mem- ber k of K with Ad(k)a0 = a0. Consequently the space p0 satisfies p0 = k∈K Ad(k)a0. (b) Any two choices of n0 are conjugate by Ad of a member of NK (a0). (c) Define W (G, A)=NK (a0)/ZK (a0). The Lie algebra of the normalizer NK (a0) is m0, and therefore W (G, A) is a finite group. (d) W (G, A) coincides with W (Σ). Reference. [K3, §VI.5]. Remarks. Already we know from the Corollary to Theorem 4.2 that any two Cartan decompositions of g0 are conjugate via Int g0. Therefore any two choices of K are conjugate in G. Conclusion (a) of the theorem says that with K fixed, any two choices of a0 are conjugate, and conclusion (b) says that with K and a0 fixed, any two choices of n0 are conjugate. Therefore any two Iwasawa decompositions are conjugate. Now let us study Cartan subalgebras and subgroups. We know that g0 always has a Cartan subalgebra. Namely if t0 is any maximal abelian subspace of m0, then h0 = a0 ⊕ t0 is a Cartan subalgebra of g0. However, Cartan subalgebras are not necessarily unique up to conjugacy, as the following example shows. 24 A. W. KNAPP R Example. The Lie algebra g0 =sl(2, ) has two Cartan subalgebras nonconju- x 0 0 y gate via Int g , namely all and all . Every Cartan subalgebra 0 0 −x −y 0 of g0 is conjugate via Int g0 to one of these. In a complex Lie algebra g, any two Cartan subalgebras are conjugate via Int g. Therefore, despite the nonconjugacy, any two Cartan subalgebras of g0 have the same dimension. This dimension is called the rank of g0. Let us mention some properties of Cartan subalgebras of g0 (see [K3, §VI.6]). Any Cartan subalgebra is conjugate via Int g0 to a θ stable Cartan subalgebra. If h0 is a θ stable Cartan subalgebra, we can decompose h0 according to g0 = k0 ⊕ p0 as h0 = t0 ⊕ a0 with t0 ⊆ k0 and a0 ⊆ p0. It is appropriate to think of t0 as the compact part of h0 and a0 as the noncompact part. Define h0 to be maximally compact if its compact part has maximal dimension among all θ stable Cartan subalgebras, or to be maximally noncompact if its noncompact part has maximal dimension. The Cartan subalgebra h0 constructed after the Iwasawa decomposition is maximally noncompact. If t0 is a maximal abelian subspace of k0, then h0 = Zg0 (t0) is maximally compact. Among θ stable Cartan subalgebras h0 of g0, the maximally noncompact ones are all conjugate via K, and the maximally compact ones are all conjugate via K. Hence the constructions in the previous paragraph yield all maximally compact and maximally noncompact θ stable Cartan subalgebras. Up to conjugacy by Int g0, there are only finitely many Cartan subalgebras of g0. In fact, any θ stable Cartan subalgebra, up to conjugacy, can be transformed into any other θ stable Cartan subalgebra by a sequence of Cayley transforms, which change a Cartan subalgebra of g0 only within a subalgebra sl(2, R). Within the sl(2, R), the change is essentially the change between the two types in the example above. The relevant sl(2, R)’s for the Cayley transforms are the ones corresponding to particular kinds of roots. By definition a Cartan subgroup of G is the centralizer in G of a Cartan subalgebra of g0. In order to analyze noncompact semisimple groups, one wants an analog of the result Theorem 3.7 in the compact case that every element is conjugate to a member of a maximal torus. For this purpose we introduce the regular elements of G. Let l be the common dimension of all Cartan subalgebras of g0, and write n −1 n j det((λ + 1)1n − Ad(x)) = λ + Dj(x)λ . j=0 We call x ∈ G regular if Dl(x) = 0. Let G be the set of all regular elements in G. Theorem 4.7. Let (h1)0,...,(hr)0 be a maximal set of nonconjugate θ stable Cartan subalgebras of g0, and let H1,...,Hr be the corresponding Cartan subgroups of G. Then ⊆ r −1 (a) G i=1 x∈G xHix . (b) each member of G lies in just one Cartan subgroup of G. Reference. [K3, Theorem 7.108]. STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 25 Remarks. By the theorem the regular elements are conjugate to members of Cartan subgroups. This fact turns out to be good enough to give an analog of the Weyl Integration Formula for noncompact semisimple groups. We omit the details. This completes our discussion of Cartan subalgebras and Cartan subgroups. We turn now to the topic of parabolic subalgebras and parabolic subgroups. The notation remains unchanged. First we introduce two subgroups M and N −. The group N − is often called N in the literature. The subgroup M of G is defined by M = ZK (a0). Its Lie algebra is m0 = Zk0 (α0), and M normalizes each restricted-root space (g0)λ. It follows from the Iwasawa decomposition (Theorem 4.5) that MAN is a closed subgroup of G. It and its conjugates in G are called minimal parabolic subgroups. ⊕ ⊕ Its Lie algebra is m0 a0 n0,aminimal parabolic subalgebra of g0. − − Let n0 = λ∈Σ+ (g0)−λ = θn0, and let N =ΘN be the corresponding analytic subgroup of G. Here is a handy integral formula used in analysis on G; for g = SL(2, R), it amounts to an arctangent substitution for passing from the circle to the line. Theorem 4.8. Write elements of G = KAN as g = κeH(g)n.Let2ρ be the sum of the members of Σ+ with multiplicities counted. Then there exists a normalization of Haar measures such that f(k) dk = f(κ(¯n))e−2ρH(¯n) dn¯ K N − for all continuous f on K that are right invariant under M. Reference. [K3, Proposition 8.46]. The next theorem gives the double-coset decomposition of G relative to the subgroup MAN. Theorem 4.9 (Bruhat decomposition). Let {w˜} be a set of representatives in K for the members w of W (G, A), and let [˜w] be the image of w˜ in W (G, A). Then G = MANwMAN˜ [˜w]∈W (G,A) disjointly. Reference. [K3, Theorem 7.40]. The existence half of the following decomposition is an immediate consequence of the global Cartan decomposition (Theorem 4.3) and the conjugacy of the various choices for a0 (Theorem 4.6). Theorem 4.10 (KAK decomposition). Every element in G has a decomposi- tion as k1ak2 with k1,k2 ∈ K and a ∈ A. In this decomposition, a is uniquely determined up to conjugation by a member of W (G, A).Ifa is fixed as exp H with H ∈ a0 and if λ(H) =0 for all λ ∈ Σ, then k1 is unique up to right multiplication by a member of M. Before considering general parabolic subalgebras and subgroups, we mention special features of the “complex case.” Suppose that the real semisimple Lie algbra 26 A. W. KNAPP Lie algebra g0 is actually complex, i.e., that there exists a linear map J : g0 → g0 such that J[X, Y ]=[JX,Y]=[X, JY ] and J 2 = −1. The corresponding group G then has an invariant complex structure and is called a complex semisimple group. Any choice of k0 is a compact real form of g0, and p0 = Jk0. The Lie algebra m0 is Ja0, and a0 ⊕ Ja0 is a complex Cartan subalgebra of the complex Lie algebra g0. Each restricted root space has real dimension 2 and is a root space for a0 ⊕ Ja0. The group M is connected, all Cartan subalgebras are complex and are conjugate, and all Cartan subgroups are connected. Returning to an arbitrary real semisimple Lie algebra g0, let us now give the definitions of general parabolic subalgebras and subgroups. A Borel subalgebra of our complex semisimple Lie algebra g is defined to be a subalgebra of the form ⊕ + h α∈∆+ gα, where h is a Cartan subalgebra and ∆ is a positive system of roots. A parabolic subalgebra of g is a subalgebra containing a Borel subalgebra. Theorem 4.11. The parabolic subalgebras containing a given Borel subalgebra may be parametrized as follows. Let Π be the set of simple roots defining the set ∆+ of positive roots that determine the Borel subalgebra. If Π is any subset of Π, then there is a parabolic subalgebra corresponding to Π , namely pΠ = h ⊕ gα ⊕ gα α∈span(Π ) other α∈∆+ = Levi subalgebra ⊕ nilpotent radical . All parabolic subalgebras containing the given Borel subalgebra are of this form. Reference. [K3, Proposition 5.90]. C Now let us consider g0. Suppose above that h =(h0) with h0 constructed from the Iwasawa decomposition and with ∆+ consistent with Σ+. Then one can show that the parabolic subalgebras of g that are complexifications are the complexifica- tions of all subalgebras of g0 containing a minimal parabolic q0 = m0 ⊕ a0 ⊕ n0. We can parametrize these by subsets of simple restricted roots as follows. The formulas look similar to those in Theorem 4.11. Let Φ be a subset of simple restricted roots. Define (qΦ)0 = m0 ⊕ a0 ⊕ (g0)λ ⊕ (g0)λ λ∈span(Φ) other λ∈Σ+ =((mΦ)0 ⊕ (aΦ)0) ⊕ (nΦ)0, ⊕ where (aΦ)0 = λ∈Φ ker λ and (mΦ)0 is the orthocomplement of (aΦ)0 in (mΦ)0 (aΦ)0. See [K3, §VII.7]. The decomposition (qΦ)0 =((mΦ)0 ⊕ (aΦ)0) ⊕ (nΦ)0 is called the Langlands decomposition of (qΦ)0. The corresponding parabolic subgroup is the normalizer QΦ = NG((qΦ)0). This is a closed subgroup of G, being a normalizer. It has a Langlands decomposition QΦ = MΦAΦNΦ, with the factors defined as follows: (MΦ)0,AΦ,NΦ are to be connected, and MΦ = M(MΦ)0. See [K3, §VII.7]. Finally we mention reductive Lie groups. Any representation theory done for the semisimple group G needs to be done also for all MΦ, but MΦ is not necessarily connected and (MΦ)0 is not necessarily semisimple. One wants a class of groups STRUCTURE THEORY OF SEMISIMPLE LIE GROUPS 27 containing interesting semisimple groups and closed under passage to the MΦ’s. Such groups are usually called reductive Lie groups. There are various definitions, depending on the author. Here is the definition of G in the Harish-Chandra class: (a) g0 is reductive, (b) G has finitely many components, (c) the analytic subgroup of G corresponding to [g0, g0] has finite center, and C (d) the action of every Ad(g)on(g0) is in Int g. These groups have a number of important properties that we state in a qualitative form. First, g0 has a Cartan involution θ. Second, G has a corresponding global Cartan decomposition. Third, the centralizer in G of any abelian θ stable subalge- bra of g0 is again in the class. Fourth, M meets every component of G. Fifth, the basic decompositions extend from the semisimple finite-center case to the reductive case. See [K3, §VII.2]. References [He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [Hu] J. E. Humphreys, Introduction to Lie Algebras and Repressentation Theory, Springer- Verlag, New York, 1972. [J] N. Jacobson, Lie Algebras, Interscience Publishers, New York, 1962; second edition, Dover Publications, New York, 1979. [K1] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, Princeton, N.J., 1986. [K2] A. W. Knapp, Lie Groups, Lie Algebras, and Cohomology, Princeton University Press, Princeton, N.J., 1988. [K3] A. W. Knapp, Lie Groups Beyond an Introduction, Birkh¨auser, Boston, 1996. [V] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall, Englewood Cliffs, N.J., 1974; second edition, Springer-Verlag, New York, 1984. [W] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer-Verlag, New York, 1972. Department of Mathematics, State University of New York, Stony Brook, New York 11794, U.S.A. E-mail address: [email protected]